Questions about root system for arbitrary Lie algebra

Question

According to the book "Introduction to Lie Algebras" by Erdmann and Wildon.

I understand that we can find root space decomposition for a semisimple Lie algebra. However, when the book goes to the definition of root system. it appears to me that root system is NOT only for semisimple Lie algebra but for any arbitrary lie algebra.

If my assertion is correct, how can I find roots for an arbitrary lie algebra if it can not be root space decomposed?

For example how can I know the roots of $\mathfrak{gl}(n,\mathbb{C})$?

Thanks in advance!

Answer 1

There is a weight space decomposition for all finite-dimensional Lie algebras $L$ over a field $K$ of characteristic zero, which later is used for the root space decomposition in the semisimple case. For this, let $H$ be a Lie subalgebra of $L$, and consider the restriction of the adjoint representation to $H$, i.e., ${\rm ad}:H\rightarrow \mathfrak{gl}(L)$, and define the generalized eigenspace $$ L_{\lambda}(h)=\{ x\in L\mid (ad(h)-\lambda id)^nx=0 \text{ for some } n\}, $$ for $h\in H$. If $K$ is algebraically closed, the Jordan decomposition of $ad(h)$ gives $$ L=L_0(h)\oplus \bigoplus_{i=1}^p L_{\lambda_i}(h), $$ where $0,\lambda_1,\ldots ,\lambda_p$ are the distinct eigenvalues of $ad(h)$. Now for each function $\alpha:H\rightarrow K$ let $$ L_{\alpha}=\bigcap_{h\in H}L_{\alpha(h)}(h). $$ Then the result is:

Theorem: Let $K$ be algebraically closed and $H\subseteq L$ be a nilpotent subalgebra. Then we have the weight space decomposition $$ L=\bigoplus_{\alpha:H\rightarrow K}L_{\alpha}, $$ such that $H\subseteq L_0$, $[L_{\alpha},L_{\beta}]\subseteq L_{\alpha+\beta}$, and $[H,L_{\alpha}]\subseteq L_{\alpha}$.